【正文】 ergraduate are just enough to bring him or her to the brink of a solution to various aspects of the problem. Yet, in each case, realworld considerations plicate the situation significantly.Fortunately these plications are not insurmountable, and the result is a very beneficial design experience. The remainder of this section looks at a few aspects of the problem which present the type of learning opportunity just described. Section discusses some of the features of a simplified mathematical model of the thermal properties of the system and how it can be easily validated experimentally. Section describes how realistic control algorithm designs can be arrived at using introductory concepts in control design. Section points out some important deficiencies of such a simplified modeling/control design process and how they can be overe through simulation. Finally, Section gives an overview of some of the microcontrollerrelated design issues which arise and learning opportunities offered. MathematicalModelLumpedelement thermal systems are described in almost any introductory linear control systems text, and just this sort of model is applicable to the slide dryer problem. Figure 4 shows a secondorder lumpedelement thermal model of the slide dryer. The state variables are the temperatures Ta of the air in the box and Tb of the box itself. The inputs to the system are the power output q(t) of the heater and the ambient temperature T165。. ma and mb are the masses of the air and the box, respectively, and Ca and Cb their specific heats. μ1 and μ2 are heat transfer coefficients from the air to the box and from the box to the external world, respectively.It’s not hard to show that the (linearized) state equationscorresponding to Figure 4 areTaking Laplace transforms of (1) and (2) and solving for Ta(s), which is the output of interest, gives the following openloop model of the thermal system:where K is a constant and D(s) is a secondorder , tz, and the coefficients of D(s) are functions of the variousparameters appearing in (1) and (2).Of course the various parameters in (1) and (2) are pletely unknown, but it’s not hard to show that, regardless of their values, D(s) has two real zeros. Therefore the main transfer function of interest (which is the one from Q(s), since we’ll assume constant ambient temperature) can be writtenMoreover, it’s not too hard to show that 1=tp1 1=tz 1=tp2, ., that the zero lies between the two poles. Both of these are excellent exercises for the student, and the result is the openloop polezero diagram of Figure 5.Obtaining a plete thermal model, then, is reduced to identifying the constant K and the three unknown time constants in (3). Four unknown parameters is quite a few, but simple experiments show that 1=tp1 _ 1=tz。1=tp2 so that tz。tp2 _ 0 are good approximations. Thus the openloop system is essentially firstorder and can therefore be written (where the subscript p1 has been dropped).Simple openloop step response experiments show that,for a wide range of initial temperatures and heat inputs, K _0:14 _=W and t _ 295 Control System DesignUsing the firstorder model of (4) for the openloop transfer function Gaq(s) and assuming for the moment that linear control of the heater power output q(t) is possible, the block diagram of Figure 6 represents the closedloop system. Td(s) is the desired, or setpoint, temperature,C(s) is the pensator transfer function, and Q(s) is the heater output in watts.Given this simple situation, introductory linear control design tools such as the root locus method can be used to arrive at a C(s) which meets the step response requirements on rise time, steadystate error, and overshoot specified in Table 1. The upshot, of course, is that a proportional controller with sufficient gain can meet all specifications. Overshoot is impossible, and increasing gains decreases both steadystate error and rise time.Unfortunately, sufficient gain to meet the specifications may require larger heat outputs than the heater is capable of producing. This was indeed the case for this system, and the result is that the rise time specification cannot be met. It is quite revealing to the student how useful such an oversimplified model, carefully arrived at, can be in determining overall performance limitations. Simulation ModelGross performance and its limitations can be determined using the simplified model of Figure 6, but there are a number of other aspects of the closedloop system whose effects on performance are not so simply modeled. Chief among these arequantization error in analogtodigital conversion of the measured temperature and the use of PWM to control the heater.Both of these are nonlinear and timevarying effects, and the only practical way to study them is through simulation (or experiment, of course).Figure 7 shows a SimulinkTM block diagram of the closedloop system which incorporates these effects. A/D converter quantization and saturation are modeled using standard Simulink quantizer and saturation blocks. Modeling PWM is more plicated and requires a custom Sfunction to represent it.This simulation model has proven particularly useful in gauging the effects of varying the basic PWM parameters and hence selecting them appropriately. (., the longer the period, the larger the temperature error PWM introduces. On the other hand, a long period is desirable to avoid excessive relay “chatter,” among other things.) PWM is often difficult for students to grasp, and the simulation model allows an exploration of its operation and effects which is quite revealing. The MicrocontrollerSimple closedloop control, keypad reading, and display control are some of the classic applications of microcontrollers, and this project incorporates all three. It is therefore an excellent allaround exercise in microcontroller applications. In addition, because the project is t